In the vertexes of a regular $ 100$-gon we place the numbers from $ 1$ to $ 100$ , in some order, every number appearing exactly once. We say that an arrangment of the numbers is happy if for every simmetry axis of the polygon, the numbers which are from one side of the axis are greater that their respective simmetrics (we don't take into consideration the numbers which are on the axis) Find all happy arrangments (If two happy arrangments are the same under the rotation they are considered as only one)

Triangle $ ABC$ is inscript in a circumference $ \Gamma$. A chord $ MN=1$ of $ \Gamma$ intersects the sides $ AB$ and $ AC$ at $ X$ and $ Y$ respectively, with $ M$, $ X$, $ Y$, $ N$ in that order in $ MN$. Let $ UV$ be the diameter of $ \Gamma$ perpendicular to $ MN$ with $ U$ and $ A$ in the same semiplane respect to $ MN$. Lines $ AV$, $ BU$ and $ CU$ cut $ MN$ in the ratios $ \frac{3}{2}$, $ \frac{4}{5}$ and $ \frac{7}{6}$ respectively (starting counting from $ M$). Find $ XY$

-Find all the functions from $ \mathbb R^+ \to \mathbb R^+$ that satisfy: $ x^2(f(x)+f(y))=(x+y)(f(yf(x)))$

Let $ d(n)$ be the number of positive divisors of the natural number $ n$. Find all $ n$ such that $ \frac {n} {d(n)}=p$ where $ p$ is a prime number Daniel

Let $ ABC$ be a triangle, $ D$, $ E$ and $ F$ the points of tangency of the incircle with sides $ BC$, $ CA$, $ AB$ respectively. Let $ P$ be the second point of intersection of $ CF$ and the incircle. If $ ABPE$ is a cyclic quadrilateral prove that $ DP$ is parellel to $ AB$

Show that in acute triangle ABC we have: $\frac{a^5+b^5+c^5}{a^4+b^4+c^4} \ge \sqrt{3}R$